Clean the torn area with acetone.
Fill in the tear with a liquid seam sealer.
Place a heavy object such as a book over the area and allow the sealer to dry.
Place an extra piece of vinyl flooring sheeting on top of the area with the tear. Look to make sure any patterns on the replacement piece line up with the torn part of the floor.
Heat up the replacement patch and the torn part of the floor with a hair dryer. The heat makes the vinyl easier to cut.
Place the replacement piece back on top of the piece to be removed. Cut through both of the pieces with a utility knife. Check again to make sure the patterns of the two pieces are the same.
Use a putty knife to pry out the torn piece of flooring.
Place some vinyl adhesive on a putty knife. Place the adhesive under the edges of the vinyl sheet that surround the piece you just removed. Apply some vinyl adhesive to the bare part of the floor.
Press the replacement patch into place and cover it with a heavy object.
Wait 24 hours and fill in the seams around the patch with a liquid seam sealer. Allow the liquid seam sealer to dry.